(1)Since , and f=0 if restricted to M, then

for all y in M.

(2)By (1), we have d(x,M) is greater or equal to sup { f(x): f in X', norm f on X' ≤ 1, f/m =0},

By Hahn-Banach Theorem, there is f satisfies the required condition,

thus the conclusion is proved.

(3)if M is dense in X, then for any element x in X, there is sequence in M converge to the element, since f in X' is continous, thus f(x)=0, for any x in X.

and the "only if" part can be easily proved by Arguement combined with Hahn-Banach Theorem which leads to contradiction.